MinT - Best Known (40, 123, s)-Nets in Base 9

Best Known (40, 123, s)-Nets in Base 9

(40, 123, 81)-Net over F₉ — Constructive and digital

Digital (40, 123, 81)-net over F₉, using

t-expansion [i] based on digital (32, 123, 81)-net over F₉, using
- net from sequence [i] based on digital (32, 80)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 32 and N(F) ≥ 81, using
    - F₄ from the tower of function fields by Bezerra and GarcÃa over F₉ [i]

(40, 123, 140)-Net over F₉ — Digital

Digital (40, 123, 140)-net over F₉, using

t-expansion [i] based on digital (39, 123, 140)-net over F₉, using
- net from sequence [i] based on digital (39, 139)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 39 and N(F) ≥ 140, using
    - a curve from the manYPoints database [i]

(40, 123, 1369)-Net in Base 9 — Upper bound on s

There is no (40, 123, 1370)-net in base 9, because

1 times m-reduction [i] would yield (40, 122, 1370)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 268 976950 409630 786794 987684 415298 416410 932528 449724 514468 935331 792682 642308 847585 954054 587315 950019 399309 062362 205777 > 9¹²² [i]